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#### Erscheinungsjahr

- 2012 (5) (entfernen)

The direction splitting approach proposed earlier in [6], aiming at the efficient solution of Navier-Stokes equations, is extended and adopted here to solve the Navier-Stokes-Brinkman equations describing incompressible flows in plain and in porous media. The resulting pressure equation is a perturbation of the
incompressibility constrained using a direction-wise factorized operator as proposed in [6]. We prove that this approach is unconditionally stable for the unsteady Navier-Stokes-Brinkman problem. We also provide numerical illustrations of the method's accuracy and efficiency.

In this paper, we propose multi-level Monte Carlo(MLMC) methods that use ensemble level mixed multiscale methods in the simulations of multi-phase flow and transport. The main idea of ensemble level multiscale methods is to construct local multiscale basis functions that can be used for any member of the ensemble. We consider two types of ensemble level mixed multiscale finite element methods, (1) the no-local-solve-online ensemble level method (NLSO) and (2) the local-solve-online ensemble level method (LSO). Both mixed multiscale methods use a number of snapshots of the permeability media to generate a multiscale basis.
As a result, in the offline stage, we construct multiple basis functions for
each coarse region where basis functions correspond to different realizations.
In the no-local-solve-online ensemble level method one uses the whole set of pre-computed basis functions to approximate the solution for an arbitrary realization. In the local-solve-online ensemble level method one uses the pre-computed functions to construct a multiscale basis for a particular realization. With this basis the solution corresponding to this
particular realization is approximated in LSO mixed MsFEM. In both approaches
the accuracy of the method is related to the number of snapshots computed based on different realizations that one uses to pre-compute a
multiscale basis. We note that LSO approaches share similarities with reduced basis methods [11, 21, 22].
In multi-level Monte Carlo methods ([14, 13]), more accurate (and expensive) forward simulations are run with fewer samples while less accurate(and inexpensive) forward simulations are run with a larger number of samples. Selecting the number of expensive and inexpensive simulations carefully, one can show that MLMC methods can provide better accuracy
at the same cost as MC methods. In our simulations, our goal is twofold. First, we would like to compare NLSO and LSO mixed MsFEMs. In particular, we show that NLSO
mixed MsFEM is more accurate compared to LSO mixed MsFEM. Further, we use both approaches in the context of MLMC to speed-up MC
calculations. We present basic aspects of the algorithm and numerical
results for coupled flow and transport in heterogeneous porous media.

The paper production is a problem with significant importance for the society
and it is a challenging topic for scientific investigations. This study is concerned
with the simulations of the pressing section of a paper machine. A two-dimensional
model is developed to account for the water flow within the pressing zone. Richards’
type equation is used to describe the flow in the unsaturated zone. The dynamic capillary
pressure–saturation relation proposed by Hassanizadeh and co-workers (Hassanizadeh
et al., 2002; Hassanizadeh, Gray, 1990, 1993a) is adopted for the paper
production process.
The mathematical model accounts for the co-existence of saturated and unsaturated
zones in a multilayer computational domain. The discretization is performed
by the MPFA-O method. The numerical experiments are carried out for parameters
which are typical for the production process. The static and dynamic capillary
pressure–saturation relations are tested to evaluate the influence of the dynamic
capillary effect.

In this work we extend the multiscale finite element method (MsFEM)
as formulated by Hou and Wu in [14] to the PDE system of linear elasticity.
The application, motivated from the multiscale analysis of highly heterogeneous
composite materials, is twofold. Resolving the heterogeneities on
the finest scale, we utilize the linear MsFEM basis for the construction of
robust coarse spaces in the context of two-level overlapping Domain Decomposition
preconditioners. We motivate and explain the construction
and present numerical results validating the approach. Under the assumption
that the material jumps are isolated, that is they occur only in the
interior of the coarse grid elements, our experiments show uniform convergence
rates independent of the contrast in the Young's modulus within the
heterogeneous material. Elsewise, if no restrictions on the position of the
high coefficient inclusions are imposed, robustness can not be guaranteed
any more. These results justify expectations to obtain coefficient-explicit
condition number bounds for the PDE system of linear elasticity similar to
existing ones for scalar elliptic PDEs as given in the work of Graham, Lechner
and Scheichl [12]. Furthermore, we numerically observe the properties
of the MsFEM coarse space for linear elasticity in an upscaling framework.
Therefore, we present experimental results showing the approximation errors
of the multiscale coarse space w.r.t. the fine-scale solution.

In this work, some model reduction approaches for performing simulations
with a pseudo-2D model of Li-ion battery are presented. A full pseudo-2D model of processes in Li-ion batteries is presented following
[3], and three methods to reduce the order of the full model are considered. These are: i) directly reduce the model order using proper
orthogonal decomposition, ii) using fractional time step discretization in order to solve the equations in decoupled way, and iii) reformulation
approaches for the diffusion in the solid phase. Combinations of above
methods are also considered. Results from numerical simulations are presented, and the efficiency and the accuracy of the model reduction approaches are discussed.